USF MSDS 601 Project: NBA Player of the Week

USF MSDS 601: Linear Regression Analysis

Project: NBA Player of the Week

Team Members

• Shirley Li (@Shirleyiscool)
• Charles Siu (@chunheisiu)

Description of Dataset

Our dataset is a combination of the following datasets with regards to NBA:

• NBA Player of the Week (1985 - 2019)
  https://www.kaggle.com/jacobbaruch/nba-player-of-the-week
  1,187 Rows, 14 Columns

• NBA Player Salary from basketball-reference.com (1991 - 2017)
  https://www.kaggle.com/whitefero/nba-player-salary-19902017
  11,837 Rows, 7 Columns

• NBA Player Salary from basketball-reference.com (2018 - 2019)
  https://web.archive.org/web/20181002194236/www.basketball-reference.com/contracts/players.html
  578 Rows, 11 Columns

• NBA Player Statistics (1985 - 2019)
  https://www.basketball-reference.com/leagues/
  18,480 Rows, 30 Columns

• NBA Yearly Summary (1985 - 2019)
  https://www.basketball-reference.com/leagues/
  35 Rows, 8 Columns

Combining the aforementioned datasets, we created a dataset in which, each row is an NBA player per season, and each column is a statistic of the player. We filtered the rows so that only the players who have both statistics and salary data for that particular season are included.

There are 9,003 Rows and 38 Columns in the dataset.

Index of the Dataset

Variables of the Dataset

Statement of Research Problems and Methods

Using the dataset, we stemmed two main research problems:

• What player statistic contributes the most to the event that the player is named Player of the Week?
  Since whether a player is named Player of the Week is a binary variable, we decided to approach this problem using the logistic regression model.

• What NBA title, including Player of the Week, has the most weight on the salary of the player?
  Since the salary of a player is a numerical variable, we decided to approach this problem using the multiple linear regression model.

For both problems, model selection was performed to find the optimal model, and model diagnosis was performed to mitigate the possible issues of heteroscedasticity, multicollinearity and autocorrelation.

Problem 1: Relationship between Player Statistics and Player of the Week

Explanatory Analysis

After extracting the relevant player statistics and Player of the Week from the dataset, we plotted the relationship between the statistics and Player of the Week using a scatter plot.

Observing the scatter plot, since Potw is a binary variable, the scatter plot did not give us a lot of useful information, apart from the differences in range of statistic values between the Potw = 0 and Potw = 1. For every statistic, the range of values seems to be smaller for Potw = 1, with the most significant variable being eFG_Prct.

This discrepancy in range is also evident in the difference in frequency between Potw = 0 and Potw = 1.

The frequency table shows that Potw = 0 accounts for 94% of the data, which is to be expected since the number of players receiving an award would always be significantly smaller than those who did not. However, we are not sure if this would effect the reliability of the models we would build in regression analysis.

We also plotted the correlation using a heatmap.

Observing the heatmap, there are evidence that multicollinearity might exist. For example, The most correlated variables are Two_P_Prct and FG_Prct, but this is to be expected since FG_Prct is derived from Two_P_Prct. Similarly, eFG_Prct is derived from FG_Prct, so the correlation is high between them. Hence, some of these variables, specifically those that have direct relationships, will need to be removed prior to regression analysis.

Meanwhile, TOV, AST and STL are highly correlated between one another. However, turnovers, assists and steals are basketball moves often performed by point guards, so there might be indirect relationships between these variables. Nonetheless, these correlations would need to be addressed in regression analysis.

Regression Analysis

As Pos is categorical variables, we first get dummies for this predictors and fit the full model using all the other players' statistics, such as Age,G,GS,MP,FG and etc.. Hence, we got the following logistic regression model as follows.

Full Model Summary

Given that there are too many variables with high correlation from the heatmap above as well as the there is warnning on multicollinearity, we decided to first use vif to remove variables with max vif values until all the remaining predictors with vif values less than 10.

Vif analysis on full model

Using a function to remove variables with max vif for every vif analysis, we got these remaining predictors with small vif values finally.

So, with these predictors, we then do the reduced model regression. The result is as follows.

Reduced Model Summary

To make sure whether reduced model is better than the full model, we do a deviance test.

Null Hypothesis: Reduced model

ΔG = ΔG(Reduced Model) - ΔG(Full Model) = 1797 - 1592 = 205

On significat level of 0.05, ΔG > \chi^2. Therefore, we should reject

Problem 2: Relationship between NBA Titles and Player Salary

Explanatory Analysis

After extracting the relevant player titles and salary from the dataset, we plotted the relationship between the titles and salary using a scatter plot.

Observing the scatter plot, since all of the variables are binary except for Year, it is difficult to interpret the relationships using the scatter plot. Looking at the Year plot, there is evidently a positive linear relationship between Year and Salary. What is also interesting about the Year graph is that, despite having a positive relationship, the range of values also increased for every season.

This change in range may have been affected by the increase in observations over the years.

From the frequency table, we can clearly see the increase in observations over the seasons. The cause of this is unknown; either there is a steady increase in players, or there is a steady increase in data collected. Nonetheless, this might be worth looking into and be cautious about during regression analysis.

We also plotted the correlation using a heatmap.

Observing the heatmap, the overall correlation seems pretty low, except for WS_Leader and MVP. This means that, except for MVP and Win Shares Leader, having one NBA title does not automatically entitled you to another. It also meant that multicollinearity is likely not an issue in regression analysis.

Regression Analysis

We first fitted the full model with variables Year, Potw, APG_Leader, MVP, PPG_Leader, RPG_Leader, Rookie, WS_Leader.

Model Summary

ANOVA Table

From the model summary and ANOVA table, it is evident that MVP and WS_Leader are not statistically significant variables based on both t-test and F-test. However, since they are highly correlated, as mentioned above, it is likely that only one of them would need to be remove.

Upon checking for non-linearity of the model, we found signs of non-linearity from the residual plot.

Hence, we performed a log transformation on Salary to attempt to correct the problem. We refitted the full model using the log-transformed data.

Model Summary

ANOVA Table

From the model summary, the log-transformed data confirmed the statistical insignificance of MVP. Meanwhile, the ANOVA table shows that Rookie and WS_Leader are also statistically insignificant. We kept that in mind for model selection.

The log-transformed data had seemingly reduced the severity of non-linearity.

Model Selection

We then proceeded to Model Selection using Adjusted R-Squared, Mallow's CP, AIC, and BIC.

Best Subset Regression Table

From this table, we can see that, the models with 5 and 6 predictors performed relatively similar. They differ in whether WS_Leader is included as a variable. It is noted above that WS_Leader might be insignificant as shown in the F-test result. We chose to regress both models and compare.

Regressing the model with 6 variables, we got the following results.

Model Summary

ANOVA Table

Regressing the model with 5 variables, we got the following results.

Model Summary

ANOVA Table

It is clear that the model Salary ~ Year + Potw + APG_Leader + PPG_Leader + RPG_Leader performed better without WS_Leader, as shown in t-test and F-test results in the model summary and ANOVA table.

Model Diagnosis

We first checked whether influential points exist in the model.

From the influence plot, it is evident that some of the observations are influential. After calculating the Cook's Distance for each observation, we found that 134 observations are influential under the 4 / n - p heuristic threshold. We decided not to remove them, since they may lead to some insights about the model.

We then check for heteroscedasticity using the Breusch-Pagan test.

Breusch-Pagan Test Results

From the p-values of LM-test and F-test in the Breusch-Pagan test, we determined that, it is unlikely that the model suffers from heteroscedasticity.

We then check for multicollinearity using both the Breusch-Godfrey test and VIF.

Breusch-Godfrey Results

VIF Test Results

From the p-values of LM-test and F-test in the Breusch-Godfrey test, as well as the VIF factors from the VIF test result, we determined that, it is unlikely that the model suffers from multicollinearity.

We finally checked for non-normality using QQ plot.

From the QQ plot, we determined that, it is unlikely that the model suffers from non-normality.

Final Model Summary

Variables

Formula